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{\bf The Horatii and the Curiatii}\footnote{Eric Rasmusen:
Professor of Business Economics
and Public Policy and Sanjay Subhedar Faculty Fellow, Indiana
University,
Kelley School of Business, BU 456,
1309 E 10th Street,
Bloomington, Indiana, 47405-1701.
Office: (812) 855-9219. Fax: 812-855-3354. Erasmuse@indiana.edu.
Php.indiana.edu/$\sim$erasmuse.} \\
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September 21, 2000 \\
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A Roman, Horatius, unwounded, is fighting the three Curiatius
brothers from Alba, each of whom is wounded. If Horatius continues
fighting, he wins with probability 0.1, and the payoffs are (10,-10)
for (Horatius, Curiatii) if he wins, and (-10,10) if he loses. With
probability $\alpha = 0.5$, Horatius is panic-stricken and runs away.
If he runs and the Curiatii do not chase him, the payoffs are (-20,
10). If he runs and the Curiatius brothers chase and kill him, the
payoffs are (-21, 20). If, however, he is not panic-stricken, but he
runs anyway and the Curiatii give chase, he is able to kill the
fastest brother first and then dispose of the other two, for payoffs
of (10,-10). Horatius is, in fact, not panic-stricken.
\hspace*{16pt} (a) With what probability $\theta$ would the
Curiatii give chase if Horatius were to run? \\
\hspace*{72pt} \underline{ {\it Answer.}} In a mixed-strategy
equilibrium,
\begin{equation} \label{e76}
\pi_h(run) = \pi_h (not\; run),
\end{equation}
so
\begin{equation} \label{e77}
\theta (10) + (1-\theta) (-20) = 0.1(10) + 0.9 (-10)
\end{equation}
and
$ {\theta^* = \frac{12}{30} =0.4}$.
\hspace*{16pt} (b) With what probability $\gamma$ does Horatius
run?\\
\hspace*{72pt} \underline{ {\it Answer.}} In a mixed-strategy
equilibrium,
\begin{equation} \label{e78}
\pi_c(chase) = \pi_c (not\; chase),
\end{equation}
so
\begin{equation} \label{e79}
\begin{array}{l}
\alpha(20) + (1-\alpha) \gamma (-10) + (1-\alpha)(1-\gamma)[ 0.1(-10)
+ 0.9 ( 10)] =\\
\alpha(10) + (1-\alpha) \gamma (10) + (1-\alpha)(1-\gamma)[ 0.1(-10) +
0.9 ( 10)],
\end{array}
\end{equation}
which reduces to
\begin{equation} \label{e80}
20 \alpha -10 \gamma + 10\alpha \gamma= 10\alpha + 10\gamma -10\alpha
\gamma,
\end{equation}
so $\gamma^* = \frac{\alpha}{2-2\alpha}$, which equals { \rm 0.5} if
$\alpha=0.5$.
\hspace*{16pt} (c) How would $\theta$ and $\gamma$ be affected
if the Curiatii falsely believed that the probability of Horatius
being panic-stricken was 1? What if they believed it was 0.9? \\
\hspace*{72pt} \underline{ {\it Answer.}} If $\alpha=1$ (or if the
Curiatii think
$\alpha=1$), there is no mixed-strategy equilibrium, because $
\pi_c(chase) > \pi_c (not\; chase)$ under all possible circumstances.
Thus, the equilibrium is $\gamma^*=1$, $\theta^*=1$: Horatius runs
and the Curiatii chase. If $\alpha =0.9$, it is still true that $
\pi_c(chase) > \pi_c (not\; chase)$ even if $\gamma=1$, so the
equilibrium is the same as if $\alpha =1$.
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